巧用“非负数”

非负数的概念及其应用在中学中占有重要地位,在各类考试和竞赛中经常碰到.如果我们在解题时,通过观察、分析而挖掘出题目中具有或隐含着的“非负数”,恰当地应用非负数的概念及其性质,巧妙地进行相应的转化,不仅可以使解题过程更加灵活、技巧简捷,而且对培养学生的思维能力和解题能力大有益处.1 非负数概念 正数和零统称为非负数,它主要包括: (1)任意实数a的绝对值,即恒有|a|≥0. (2)实数a的偶次幂。

Matrix dimensionality reduction for mining typical user profiles

Recently clustering techniques have been used to automatically discover typical user profiles. In general, it is a challenging problem to design effective similarity measure between the session vectors which are usually high-dimensional and sparse. Two approaches for mining typical user profiles, based on matrix dimensionality reduction, are presented. In these approaches, non-negative matrix factorization is applied to reduce dimensionality of the session-URL matrix, and the projecting vectors of the user-session vectors are clustered into typical user-session profiles using the spherical k -means algorithm. The results show that two algorithms are successful in mining many typical user profiles in the user sessions.

A Novel Resource Allocation Algorithm based on Downlink and Uplink Decouple Access Scheme in Heterogeneous Networks

Traditional cellular network requires that a user equipment(UE) should associate to the same base station(BS) in both the downlink(DL) and the uplink(UL). Based on dual connectivity(DC) introduced in LTE-Advanced R12, DL/UL decouple access scheme has been proposed, which is especially suitable for heterogeneous networks(Het Nets). This paper is the pioneer to take the DL/UL decouple access scheme into consideration and develop a novel resource allocation algorithm in a two-tier Het Net to improve the total system throughput in the UL and ease the load imbalance between macro base stations(MBSs) and pico base stations(PBSs). A model is formulated as a nonlinear integer programming, and the proposed algorithm is a sub-optimal algorithm based on the graph theory. First, an undirected and weighted interference graph is obtained. Next, the users are grouped to let users with large mutual interferences to be assigned to different clusters. Then, the users in different clusters are allocated to different resource blocks(RBs) by using the Hungarian algorithm. Simulation results show that the proposed algorithm can provide great promotions for both the total system throughput and the average cell edge user throughput and successfully ease the load imbalance between MBSs and PBSs.

Daily heart rate variability of Paralympic gold medallist swimmers： A 17-week investigation

Objectives： Heart rate variability （HRV） can be a simple, non-invasive method of gauging cardiac autonomic nervous system fluctuations across periodised training workloads and taper in elite athlete populations. The purpose of these three case studies was to examine daily cardiac autonomic variations in Paralympic athletes leading in to the Paralympic games. Methods： Three Paralympie gold medallist swimmers were monitored daily for their resting HRV over a 17-week monitoring period leading up to the Paralympic games. Specific time- and frequency-domain measures, along with non-linear indices of HRV were calculated for all analyses. All HRV data were analysed individually using daily values, weekly average values, and average values for rest and training phases. Results： A significant difference in HRV was seen for all variables between athlete 1 and athletes 2 and 3 （amputee disabilities） during the entire monitoring period. Conclusion： Despite minimal long-term changes, both swimming classification and disability type significantly influence HRV during athlete monitoring. An increased understanding of individual responses to training, travel, and other outside influences affecting Paralympic athletes could potentially lead to improved management and monitoring of training workloads for enhanced nerformance.

On universal sums of polygonal numbers

For m = 3, 4,..., the polygonal numbers of order m are given by pm(n) =(m- 2) n2 + n(n= 0, 1, 2,...). For positive integers a, b, c and i, j, k 3 with max{i, j, k} 5, we call the triple(api, bpj, cpk)universal if for any n = 0, 1, 2,..., there are nonnegative integers x, y, z such that n = api(x) + bpj(y)+ cpk(z). We show that there are only 95 candidates for universal triples(two of which are(p4, p5, p6) and(p3, p4, p27)), and conjecture that they are indeed universal triples. For many triples(api, bpj, cpk)(including(p3, 4p4, p5),(p4, p5, p6) and(p4, p4, p5)), we prove that any nonnegative integer can be written in the form api(x) + bpj(y) + cpk(z) with x, y, z ∈ Z. We also show some related new results on ternary quadratic forms,one of which states that any nonnegative integer n ≡ 1(mod 6) can be written in the form x2+ 3y2+ 24z2 with x, y, z ∈ Z. In addition, we pose several related conjectures one of which states that for any m = 3, 4,...each natural number can be expressed as pm+1(x1) + pm+2(x2) + pm+3(x3) + r with x1, x2, x3 ∈ {0, 1, 2,...}and r ∈ {0,..., m- 3}.

Various characterizations of product Hardy spaces associated to Schrdinger operators

For every i = 1, 2, we let Li =-?ni+ Vi be a Schr¨odinger operator on Rni in which Vi∈ L1loc(Rni)is a non-negative function on Rni. We obtain some characterizations for functions in the product Hardy space H1L1,L2(Rn1 × Rn2) associated to L1 and L2 by using different norms on distinct variables.

DOUBLE Φ-FUNCTION INEQUALITY FOR NONNEGATIVE SUBMARTINGALES

The authors establish a kind of inequalities for nonnegative submartingales which depend on two functions ψ and ψ, and obtain the equivalent conditions for Φ and ψ such that this kind of inequalities holds. In the caseΦ=ψ ε△2, it is proved that this necessary and sufficient condition is equivalent to qΦ >1.

Every 3-connected {K_(1,3),N_(3,3,3)}-free graph is Hamiltonian

For non-negative integers i,j and k,let N i,j,k be the graph obtained by identifying end vertices of three disjoint paths of lengths i,j and k to the vertices of a triangle.In this paper,we prove that every 3-connected {K1,3,N3,3,3 }-free graph is Hamiltonian.This result is sharp in the sense that for any integer i>3,there exist infinitely many 3-connected {K1,3,Ni,3,3 }-free non-Hamiltonian graphs.

bmo _ρ(ω) Spaces and Riesz transforms associated to Schrdinger operators

We introduce the BMO-type space bmo ρ（w） and establish the duality between h^1ρ（ω） and bmo ρ（ω）,where ω∈A1^ρ∞（R^n） and ω＇s locally behave as Muckenhoupt＇s weights but actually include them. We also give the Fefferman-Stein type decomposition of bmop（ω） with respect to Riesz transforms associated to Schrodinger operator L,where L=-△＋V is a SchrSdinger operator on R^2 （n≥3） and V is a non-negative function satisfying the reverse HSlder inequality.

On the values of representation functions

For a set A of nonnegative integers, the representation functions R2(A,n) and R3(A,n) are defined as the numbers of solutions to the equation n = a + a′ with a,a′∈ A, a < a′ and a a′, respectively. Let N be the set of nonnegative integers. Given n0 > 0, it is known that there exist A,A′■ N such that R2(A′,n) = R2(N \ A′,n) and R3(A,n) = R3(N \ A,n) for all n n0. We obtain several related results. For example, we prove that: If A ■ N such that R3(A,n) = R3(N \ A,n) for all n n0, then (1) for any n n0 we have R3(A,n) = R3(N \ A,n) > c1n - c2, where c1,c2 are two positive constants depending only on n0; (2) for any α < 116, the set of integers n with R3(A,n) > αn has the density one. The answers to the four problems in Chen-Tang (2009) are affirmative. We also pose two open problems for further research.

Wavelet estimations for density derivatives

Donoho et al. in 1996 have made almost perfect achievements in wavelet estimation for a density function f in Besov spaces Bsr,q(R). Motivated by their work, we define new linear and nonlinear wavelet estimators flin,nm, fnonn,m for density derivatives f(m). It turns out that the linear estimation E(‖flinn,m-f(m)‖p) for f(m) ∈ Bsr,q(R) attains the optimal when r≥ p, and the nonlinear one E(‖fnonn,m-f(m)‖p) does the same if r≤p/2(s+m)+1 . In addition, our method is applied to Sobolev spaces with non-negative integer exponents as well.

Chaos and null systems

A dynamical system is called a null system, if the topological sequence entropy along any strictly increasing sequence of non-negative integers is 0. Let 0≦p≦q≦1. A dynamical system is Dqp chaotic, if there is an uncountable subset in which any two different points have trajectory approaching time set with lower density p and upper density q. In this paper, we show that there is a null system which is also D3/41/4 chaotic.

Dynamics of a Rational Difference Equation

The authors investigate the global behavior of the solutions of the difference equation xn＋1=axn-1xn-k/bxn-p＋cxn-q,n=0,1,…where the initial conditions x-r, x-r＋1, x-r＋2,… , x0 are arbitrary positive real numbers, r = max{l, k,p, q） is a nonnegative integer and a, b, c are positive constants. Some special cases of this equation are also studied in this paper.

Positivity of Fock Toeplitz Operators via the Berezin Transform

Abstract This paper deals with the relationship between the positivity of the Fock Toeplitz operators and their Berezin transforms. The author considers the special case of the bounded radial function φ（z）=α ＋ be-α｜z｜2 ＋ ce-β｜z｜2, where a, b, c are real numbers and α,β are positive numbers. For this type of φ, one can choose these parameters such that the Berezin transform of is a nonnegative function on the complex plane, but the corresponding Toeplitz operator Tφ is not positive on the Fock space.

The invariance principle for fractionally integrated processes with strong near-epoch dependent innovations

In this paper, we show the invariance principle for the partial sum processes of fractionally integrated processes, otherwise known as I(d + m) processes, where |d| < 1/2 and m is a nonnegative integer, with strong near-epoch dependent innovations. The results are applied to the test of unit root. The conditions given improve previous results in the literature concerning fractionally integrated processes.

On infinite additive complements

Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. For a sequence T of non-negative integers, let T（x） be the number of terms of T not exceeding x. In 1994, Sarkozy and Szemer′edi confirmed a conjecture of Danzer by proving that, for infinite additive complements A and B, if lim sup A（x）B（x）/x 1, then A（x）B（x）-x → ＋∞ as x → ＋∞. In this paper, it is proved that, if A and B are infinite additive complements with lim sup A（x）B（x）/x〈（√4 ＋ 2）/7 = 1.093 …, then（A（x）B（x）-x）/min{A（x）, B（x）} → ＋∞ as x → ＋∞.

A property of Pisot numbers and Fourier transforms of self-similar measures

For any Pisot number β it is known that the set F（β） ={t ： limn→∞‖tβn‖ = 0} is countable, where ‖α‖ is the distance between a real number a and the set of integers. In this paper it is proved that every member in this set is of the form cβn, where n is a nonnegative integer and e is determined by a linear system of equations. Furthermore, for some self-similar measures μ associated with β, the limit at infinity of the Fourier transforms limn→μ（tβn）≠0 if and only if t is in a certain subset of F（β）. This generalizes a similar result of Huang and Strichartz.

An Inverse Problem of Identifying the Radiative Coefficient in a Degenerate Parabolic Equation

The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.

Increasing Powers in a Degenerate Parabolic Logistic Equation

The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem tu- △u=au-b（x）up in Ω×R＋,u（0）=u0,u（t ）｜ Ω=0, as p→ ＋∞, where Ω is a bounded domain, and b（x） is a nonnegative function. The authors deduce that the limiting configuration solves a parabolic obstacle problem, and afterwards fully describe its long time behavior.

The Auslander-type condition of triangular matrix rings

Let R be a left and right Noetherian ring and n, k be any non-negative integers. R is said to satisfy the Auslander-type condition Gn（k） if the right fiat dimension of the （i ＋ 1）-th term in a minimal injective resolution of RR is at most i ＋ k for any 0 ≤ i ≤ n - 1. In this paper, we prove that R is Gn（k） if and only if so is a lower triangular matrix ring of any degree t over R.